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Theorem opnnei 22844
Description: A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)
Assertion
Ref Expression
opnnei (𝐽 ∈ Top β†’ (𝑆 ∈ 𝐽 ↔ βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯})))
Distinct variable groups:   π‘₯,𝐽   π‘₯,𝑆
Proof of Theorem opnnei
StepHypRef Expression
1 0opn 22626 . . . . 5 (𝐽 ∈ Top β†’ βˆ… ∈ 𝐽)
21adantr 479 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 = βˆ…) β†’ βˆ… ∈ 𝐽)
3 eleq1 2819 . . . . 5 (𝑆 = βˆ… β†’ (𝑆 ∈ 𝐽 ↔ βˆ… ∈ 𝐽))
43adantl 480 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 = βˆ…) β†’ (𝑆 ∈ 𝐽 ↔ βˆ… ∈ 𝐽))
52, 4mpbird 256 . . 3 ((𝐽 ∈ Top ∧ 𝑆 = βˆ…) β†’ 𝑆 ∈ 𝐽)
6 rzal 4507 . . . 4 (𝑆 = βˆ… β†’ βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}))
76adantl 480 . . 3 ((𝐽 ∈ Top ∧ 𝑆 = βˆ…) β†’ βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}))
85, 72thd 264 . 2 ((𝐽 ∈ Top ∧ 𝑆 = βˆ…) β†’ (𝑆 ∈ 𝐽 ↔ βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯})))
9 opnneip 22843 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ π‘₯ ∈ 𝑆) β†’ 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}))
1093expia 1119 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) β†’ (π‘₯ ∈ 𝑆 β†’ 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯})))
1110ralrimiv 3143 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) β†’ βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}))
1211ex 411 . . . 4 (𝐽 ∈ Top β†’ (𝑆 ∈ 𝐽 β†’ βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯})))
1312adantr 479 . . 3 ((𝐽 ∈ Top ∧ Β¬ 𝑆 = βˆ…) β†’ (𝑆 ∈ 𝐽 β†’ βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯})))
14 df-ne 2939 . . . . . 6 (𝑆 β‰  βˆ… ↔ Β¬ 𝑆 = βˆ…)
15 r19.2z 4493 . . . . . . 7 ((𝑆 β‰  βˆ… ∧ βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯})) β†’ βˆƒπ‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}))
1615ex 411 . . . . . 6 (𝑆 β‰  βˆ… β†’ (βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ βˆƒπ‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯})))
1714, 16sylbir 234 . . . . 5 (Β¬ 𝑆 = βˆ… β†’ (βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ βˆƒπ‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯})))
18 eqid 2730 . . . . . . . 8 βˆͺ 𝐽 = βˆͺ 𝐽
1918neii1 22830 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯})) β†’ 𝑆 βŠ† βˆͺ 𝐽)
2019ex 411 . . . . . 6 (𝐽 ∈ Top β†’ (𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ 𝑆 βŠ† βˆͺ 𝐽))
2120rexlimdvw 3158 . . . . 5 (𝐽 ∈ Top β†’ (βˆƒπ‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ 𝑆 βŠ† βˆͺ 𝐽))
2217, 21sylan9r 507 . . . 4 ((𝐽 ∈ Top ∧ Β¬ 𝑆 = βˆ…) β†’ (βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ 𝑆 βŠ† βˆͺ 𝐽))
2318ntrss2 22781 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† 𝑆)
2423adantr 479 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) ∧ βˆ€π‘₯ ∈ 𝑆 {π‘₯} βŠ† ((intβ€˜π½)β€˜π‘†)) β†’ ((intβ€˜π½)β€˜π‘†) βŠ† 𝑆)
25 vex 3476 . . . . . . . . . . . . 13 π‘₯ ∈ V
2625snss 4788 . . . . . . . . . . . 12 (π‘₯ ∈ ((intβ€˜π½)β€˜π‘†) ↔ {π‘₯} βŠ† ((intβ€˜π½)β€˜π‘†))
2726ralbii 3091 . . . . . . . . . . 11 (βˆ€π‘₯ ∈ 𝑆 π‘₯ ∈ ((intβ€˜π½)β€˜π‘†) ↔ βˆ€π‘₯ ∈ 𝑆 {π‘₯} βŠ† ((intβ€˜π½)β€˜π‘†))
28 dfss3 3969 . . . . . . . . . . . . 13 (𝑆 βŠ† ((intβ€˜π½)β€˜π‘†) ↔ βˆ€π‘₯ ∈ 𝑆 π‘₯ ∈ ((intβ€˜π½)β€˜π‘†))
2928biimpri 227 . . . . . . . . . . . 12 (βˆ€π‘₯ ∈ 𝑆 π‘₯ ∈ ((intβ€˜π½)β€˜π‘†) β†’ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘†))
3029adantl 480 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) ∧ βˆ€π‘₯ ∈ 𝑆 π‘₯ ∈ ((intβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘†))
3127, 30sylan2br 593 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) ∧ βˆ€π‘₯ ∈ 𝑆 {π‘₯} βŠ† ((intβ€˜π½)β€˜π‘†)) β†’ 𝑆 βŠ† ((intβ€˜π½)β€˜π‘†))
3224, 31eqssd 3998 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) ∧ βˆ€π‘₯ ∈ 𝑆 {π‘₯} βŠ† ((intβ€˜π½)β€˜π‘†)) β†’ ((intβ€˜π½)β€˜π‘†) = 𝑆)
3332ex 411 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ (βˆ€π‘₯ ∈ 𝑆 {π‘₯} βŠ† ((intβ€˜π½)β€˜π‘†) β†’ ((intβ€˜π½)β€˜π‘†) = 𝑆))
3425snss 4788 . . . . . . . . . . . 12 (π‘₯ ∈ 𝑆 ↔ {π‘₯} βŠ† 𝑆)
35 sstr2 3988 . . . . . . . . . . . . . 14 ({π‘₯} βŠ† 𝑆 β†’ (𝑆 βŠ† βˆͺ 𝐽 β†’ {π‘₯} βŠ† βˆͺ 𝐽))
3635com12 32 . . . . . . . . . . . . 13 (𝑆 βŠ† βˆͺ 𝐽 β†’ ({π‘₯} βŠ† 𝑆 β†’ {π‘₯} βŠ† βˆͺ 𝐽))
3736adantl 480 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ({π‘₯} βŠ† 𝑆 β†’ {π‘₯} βŠ† βˆͺ 𝐽))
3834, 37biimtrid 241 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ (π‘₯ ∈ 𝑆 β†’ {π‘₯} βŠ† βˆͺ 𝐽))
3938imp 405 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) ∧ π‘₯ ∈ 𝑆) β†’ {π‘₯} βŠ† βˆͺ 𝐽)
4018neiint 22828 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ {π‘₯} βŠ† βˆͺ 𝐽 ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ (𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) ↔ {π‘₯} βŠ† ((intβ€˜π½)β€˜π‘†)))
41403com23 1124 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽 ∧ {π‘₯} βŠ† βˆͺ 𝐽) β†’ (𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) ↔ {π‘₯} βŠ† ((intβ€˜π½)β€˜π‘†)))
42413expa 1116 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) ∧ {π‘₯} βŠ† βˆͺ 𝐽) β†’ (𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) ↔ {π‘₯} βŠ† ((intβ€˜π½)β€˜π‘†)))
4339, 42syldan 589 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) ∧ π‘₯ ∈ 𝑆) β†’ (𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) ↔ {π‘₯} βŠ† ((intβ€˜π½)β€˜π‘†)))
4443ralbidva 3173 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ (βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) ↔ βˆ€π‘₯ ∈ 𝑆 {π‘₯} βŠ† ((intβ€˜π½)β€˜π‘†)))
4518isopn3 22790 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ (𝑆 ∈ 𝐽 ↔ ((intβ€˜π½)β€˜π‘†) = 𝑆))
4633, 44, 453imtr4d 293 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ (βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ 𝑆 ∈ 𝐽))
4746ex 411 . . . . . 6 (𝐽 ∈ Top β†’ (𝑆 βŠ† βˆͺ 𝐽 β†’ (βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ 𝑆 ∈ 𝐽)))
4847com23 86 . . . . 5 (𝐽 ∈ Top β†’ (βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ (𝑆 βŠ† βˆͺ 𝐽 β†’ 𝑆 ∈ 𝐽)))
4948adantr 479 . . . 4 ((𝐽 ∈ Top ∧ Β¬ 𝑆 = βˆ…) β†’ (βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ (𝑆 βŠ† βˆͺ 𝐽 β†’ 𝑆 ∈ 𝐽)))
5022, 49mpdd 43 . . 3 ((𝐽 ∈ Top ∧ Β¬ 𝑆 = βˆ…) β†’ (βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯}) β†’ 𝑆 ∈ 𝐽))
5113, 50impbid 211 . 2 ((𝐽 ∈ Top ∧ Β¬ 𝑆 = βˆ…) β†’ (𝑆 ∈ 𝐽 ↔ βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯})))
528, 51pm2.61dan 809 1 (𝐽 ∈ Top β†’ (𝑆 ∈ 𝐽 ↔ βˆ€π‘₯ ∈ 𝑆 𝑆 ∈ ((neiβ€˜π½)β€˜{π‘₯})))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068   βŠ† wss 3947  βˆ…c0 4321  {csn 4627  βˆͺ cuni 4907  β€˜cfv 6542  Topctop 22615  intcnt 22741  neicnei 22821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-top 22616  df-ntr 22744  df-nei 22822
This theorem is referenced by:  neiptopreu  22857  flimcf  23706
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